Apollo has a persisted queries option. And there's a PR open on relay modern for it. At least for queries, both of these let's you store the query itself on the server then use a GET request with a query hash and variables.

We use this at First Look Media on some of our sites.

Yes, we do [1]

We use GET for queries and POST for mutations. Though we are not caching our GETs atm.

[1] https://functional.works-hub.com